Abstraction in Context: Epistemic Actions

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 128.252.67.66 on Tue,We propose an approach to the theoretical and empirical identification processes of abstraction in context. Although our outlook is theoretical, our thinking about abstraction emerges from the analysis of interview data. We consider abstraction an activity of vertically reorganizing previously constructed mathematics into a new mathematical structure. We use the term activity to emphasize that abstraction is a process with a history; it may capitalize on tools and other artifacts, and it occurs in a particular social setting. We present the core of a model for the genesis of abstraction. The principal components of the model are three dynamically nested epistemic actions: constructing, recognizing, and building-with. To study abstraction is to identify these epistemic actions of students participating in an activity of abstraction.Abstraction has been the focus of extensive interest in several domains, including mathematics education. Many researchers have taken a predominantly theoretical stance and have described abstraction as some type of decontextualization. In this

Teacher Interventions Aimed at Mathematical Level Raising During Collaborative Learning

“…Dreyfus and his colleagues (Kieran and Dreyfus, 1998;Hershkowitz et al, 2001) have also studied these processes. They refer to level raising as 'abstraction in context' and 'an activity of vertically reorganising previously constructed mathematics into a new mathematical structure'.…”

What is the meaning of “meaning”? A case study from graphing

“…That is, explanation, which requires a critical distancing from practical understanding, can in fact develop practical understanding in the direction of more theoretical understanding. This development of theoretical understanding out of praxis is the kind of "abstraction" of graphical knowledge that others have advocated (Hershkowitz et al, 2001). This form of abstraction never severs the tie between the concrete object and how it appears to the subject, but conceives of the two as poles joined in a dialectical unit.…”

“…One recent study of graphing made substantial progress toward a more comprehensive perspective of meaning by situating graphs and graphing in practical activity whereby graphs, as artifacts, are dialectically linked with the way they appear to the acting subject (Hershkowitz et al, 2001). Their study, in which a student worked with three population-time graphs, however falls short by focusing on mental (epistemic) actions: the researchers did not address, for example, how our understanding of up-down in a graph is a metaphoric extension of our bodily experience in the world (Johnson, 1987) marked by a particular orientation of our bodies with respect to the gravitational field or how the left to right orientation of temporal development is mediated by the way a particular language is oriented (left to right, right to left, top to bottom).…”

“…This problem is centrally addressed by activity theory. Activity theory places primacy on human praxis (the lived experience of engagement in activity) as the analytic site of cognition (Leont'ev, 1978) and has been successfully used to frame knowing and learning related to graphs in schools (Hershkowitz, Schwarz, & Dreyfus, 2001) and in the workplace (Roth, 2003;Williams, Boreham, & Wake, 2001). The fundamental recognition of activity theorists is the fact that in activity, the relationship between the subject and object of activity is direct but mediated by a series of situational factors, including the means of production (tools, materials), division of labor, the relevant community (of practice), and its traditions and rules (Engeström, 1999).…”

Mathematics, PISA, and culture: An unpredictable relationship

“…Vertical mathematizing reflects the sorts of activities students undertake when working within the mathematical system itself (Barnes 2005;Van den HeuvelPanhuizen 2003). It represents a journey a learner may take within the mathematical world of symbols (Freudenthal (1991) and plays a major role in abstraction, as mathematical entities are structured, organized or developed into new entities, typically more abstract than the original (Hershkowitz et al 2001). Through this process of progressive mathematization, whereby students pass from informal solutions, through some level of schematization, to an awareness of the general principles (Van den Heuvel-Panhuizen 2003), students reinvent mathematics (Gravemeijer 1998(Gravemeijer , 2004).…”